In mathematics, the Teichmüller–Tukey lemma (sometimes named just Tukey's lemma), named after John Tukey and Oswald Teichmüller, is a lemma that states that every nonempty collection of finite character has a maximal element with respect to inclusion. Over Zermelo–Fraenkel set theory, the Teichmüller–Tukey lemma is equivalent to the axiom of choice, and therefore to the well-ordering theorem, Zorn's lemma, and the Hausdorff maximal principle.

Definitions

A family of sets \mathcal{F} is of finite character provided it has the following properties:

Statement of the lemma

Let Z be a set and let. If \mathcal{F} is of finite character and, then there is a maximal (according to the inclusion relation) such that.

Applications

In linear algebra, the lemma may be used to show the existence of a basis. Let V be a vector space. Consider the collection \mathcal{F} of linearly independent sets of vectors. This is a collection of finite character. Thus, a maximal set exists, which must then span V and be a basis for V.